Okay, well this was a question on one of my recent tests:
How many terms do you have to use to estimate the sum from n = 0 to n = infinity of
(-e/pi)^n with an error of less than .001?
Alternating series remainder theorem:
For an alternating series.
The absolute value of (S - S(n) = (Rn) = (error) is less than or equal to a(n+1)
The Attempt at a Solution
My solution: Using the alternating series remainder theorem, says that absolute value of (S-Sn)= (error) is less than or equal to a(n+1)
I find that a(48) is the first term less than .001 it is proximately equal to .000961. This means that my series must include all terms up to a(47) to have an error that is guaranteed to be less than .001. Since the series starts at 0 this means my total sum = a(0)+a(1)+a(2)+...+a(45)+a(46)+a(47). This gives me a total of 48 terms and the answer to the question is 48.
My math teacher claims that the answer is 47 total terms. I think that the answer cannot possibly be 47 terms. This means that your highest term is a(46). So again according to the alternating series remainder theorem:
error < or = a(n+1) or a(47). a(47) is a proximately equal to .00111
.00111 is not less than .001 so you are not guaranteed that your error is actually less than .001, it only has to be less than .00111.
Who is correct? Is the answer 48. Or am I just missing something and the answer is 47?